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Chapter 4: Problem 20

What is the angle between the hour hand and the minute hand on a clock at \(11: 17 ?\)

Short Answer

Expert verified
At 11:17, the angle between the hour and minute hand on a clock is approximately \(153.5^{\circ}\).

Step by step solution

01

Calculate the position of the hour hand

To find the position of the hour hand at 11:17, we can use the fact that the hour hand moves 360 degrees in 12 hours. At 11 hours, the hour hand has moved 330 degrees (11 * 30 degrees, since each hour takes up 30 degrees on the clock). To account for the extra 17 minutes, we can calculate the additional angle moved by the hour hand during these minutes: \(\frac{17}{60} \times 30\) degrees. Adding this to the 330 degrees, we get the position of the hour hand.
02

Calculate the position of the minute hand

To find the position of the minute hand at 11:17, we can use the fact that the minute hand moves 360 degrees in 60 minutes. At 17 minutes, the minute hand has moved 102 degrees (17 * 6 degrees, since each minute takes up 6 degrees on the clock).
03

Find the angle between the hour and minute hand

Now that we have the position of both the hour and minute hands, we can calculate the angle between them by finding the absolute difference between their positions. Calculate the absolute difference between the hour and minute hand angles.
04

Ensure the calculated angle is the smaller angle

The result from Step 3 may give us the larger angle between the two hands. To ensure we have the smaller angle, check if the result is greater than 180 degrees. If it is, subtract it from 360 degrees to find the smaller angle. Using the steps above, we can now calculate the angle between the hour and minute hand at 11:17.

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Most popular questions from this chapter

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\tan \frac{\pi}{8}\)

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